Summary of Computing the Bias Of Constant-step Stochastic Approximation with Markovian Noise, by Sebastian Allmeier et al.
Computing the Bias of Constant-step Stochastic Approximation with Markovian Noise
by Sebastian Allmeier, Nicolas Gast
First submitted to arxiv on: 23 May 2024
Categories
- Main: Machine Learning (stat.ML)
- Secondary: Machine Learning (cs.LG); Optimization and Control (math.OC)
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| Summary difficulty | Written by | Summary |
|---|---|---|
| High | Paper authors | High Difficulty Summary Read the original abstract here |
| Medium | GrooveSquid.com (original content) | Medium Difficulty Summary This paper explores stochastic approximation algorithms with Markovian noise and constant step-size α. The authors develop a method based on infinitesimal generator comparisons to analyze the algorithm’s bias, which is the expected difference between θn (the value at iteration n) and θ∗ (the unique equilibrium of the corresponding ODE). They show that under some smoothness conditions, this bias is of order O(α). Furthermore, they demonstrate that the time-averaged bias is equal to αV + O(α^2), where V is a constant characterized by a Lyapunov equation. The authors also illustrate how to combine this with Richardson-Romberg extrapolation to derive an iterative scheme with a bias of order O(α^2). |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary This paper studies special kinds of math problems called stochastic approximation algorithms. These algorithms try to find the answer to a question by making small changes and then averaging them out. The researchers in this paper want to know how accurate these algorithms are, so they develop a new way to understand why some answers might be different from others. They show that with some special conditions, the difference between an algorithm’s answer and the correct answer is very small. This research could help us make better algorithms for solving math problems. |
